# Field-theoretic approach to fluctuation effects in neural networks

## Abstract

A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification ofmore »

- Authors:

- NIH/NIDDK/LBM, Building 12A Room 4007, MSC 5621, Bethesda, Maryland 20892 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 21072435

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics; Journal Volume: 75; Journal Issue: 5; Other Information: DOI: 10.1103/PhysRevE.75.051919; (c) 2007 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND TECHNOLOGY; CORRECTIONS; EXPANSION; FLUCTUATIONS; MARKOV PROCESS; MEAN-FIELD THEORY; NEURAL NETWORKS; NONLINEAR PROBLEMS; PHASE TRANSFORMATIONS

### Citation Formats

```
Buice, Michael A., Cowan, Jack D., and Mathematics Department, University of Chicago, Chicago, Illinois 60637.
```*Field-theoretic approach to fluctuation effects in neural networks*. United States: N. p., 2007.
Web. doi:10.1103/PHYSREVE.75.051919.

```
Buice, Michael A., Cowan, Jack D., & Mathematics Department, University of Chicago, Chicago, Illinois 60637.
```*Field-theoretic approach to fluctuation effects in neural networks*. United States. doi:10.1103/PHYSREVE.75.051919.

```
Buice, Michael A., Cowan, Jack D., and Mathematics Department, University of Chicago, Chicago, Illinois 60637. Tue .
"Field-theoretic approach to fluctuation effects in neural networks". United States.
doi:10.1103/PHYSREVE.75.051919.
```

```
@article{osti_21072435,
```

title = {Field-theoretic approach to fluctuation effects in neural networks},

author = {Buice, Michael A. and Cowan, Jack D. and Mathematics Department, University of Chicago, Chicago, Illinois 60637},

abstractNote = {A well-defined stochastic theory for neural activity, which permits the calculation of arbitrary statistical moments and equations governing them, is a potentially valuable tool for theoretical neuroscience. We produce such a theory by analyzing the dynamics of neural activity using field theoretic methods for nonequilibrium statistical processes. Assuming that neural network activity is Markovian, we construct the effective spike model, which describes both neural fluctuations and response. This analysis leads to a systematic expansion of corrections to mean field theory, which for the effective spike model is a simple version of the Wilson-Cowan equation. We argue that neural activity governed by this model exhibits a dynamical phase transition which is in the universality class of directed percolation. More general models (which may incorporate refractoriness) can exhibit other universality classes, such as dynamic isotropic percolation. Because of the extremely high connectivity in typical networks, it is expected that higher-order terms in the systematic expansion are small for experimentally accessible measurements, and thus, consistent with measurements in neocortical slice preparations, we expect mean field exponents for the transition. We provide a quantitative criterion for the relative magnitude of each term in the systematic expansion, analogous to the Ginsburg criterion. Experimental identification of dynamic universality classes in vivo is an outstanding and important question for neuroscience.},

doi = {10.1103/PHYSREVE.75.051919},

journal = {Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics},

number = 5,

volume = 75,

place = {United States},

year = {Tue May 15 00:00:00 EDT 2007},

month = {Tue May 15 00:00:00 EDT 2007}

}